in \({K}/B\) . The Čech nerve \(\check{C}(Y)\)  can of course be
defined for any map \(\pi \colon Y\rightarrow M\) .
It is a well known fact (essentially going back to [1]})
that if \(\pi \)  admits local sections and \(M\)  is paracompact then the
map \(|\check{C}(Y)|\rightarrow M\)  is a homotopy equivalence (in fact the
hypothesis on \(M\)  can be removed and one can show, at least when \(B\)  is a point,
that the map is a weak homotopy equivalence — see [2]}).
We have the following Lemma.
